A countable model of set theory can internally assert that an uncountable set exists. There is no contradiction because “there is a bijection” ranges only over functions in the model. An external enumeration of its domain need not itself be an element of that domain. Löwenheim-Skolem theorem controls external cardinality; Cantor's theorem remains valid for the sets and functions that the model contains.
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Skolem's paradox is a result in set theory and mathematical logic that highlights a tension between the concepts of countable and uncountable sets, particularly in the context of first-order logic. The paradox arises from the work of Norwegian mathematician Thoralf Skolem in the early 20th century.